1958 AMC 12 Problem 44

Represent the comparisons as propositions and distinguish each implication from its converse

The second premise’s hypothesis is incompatible with the first premise’s conclusion, but that alone does not chain the implications

Let $P$ mean $a>b,$ $Q$ mean $c>d,$ $S$ mean $c<d,$ and $R$ mean $e>f.$ The premises are $$P\Rightarrow Q,\qquad S\Rightarrow R.$$ The statements $Q$ and $S$ cannot both hold. The contrapositives are $\neg Q\Rightarrow\neg P$ and $\neg R\Rightarrow\neg S.$ None of the four proposed implications follows. For example, knowing $P$ gives $Q$ and hence $\neg S,$ but says nothing about $R;$ knowing $\neg R$ gives $\neg S,$ not $P.$

Therefore, the correct answer is E.